Part of a proof for the existence of a Borel isomorphism of the unit line to a subset of $\{0,2\}^{\mathbb{N}}$

Question

Let $M=\{0,1\}^{\mathbb{N}}$ and $E\subset M$ such that $x(n)=1$ for all $n$ or $x(n)=0$ for infinitely many $n$. For any $x\in M$ we write $$\tau(x)=\sum_{n=1}^\infty \frac{x(n)}{2^n}.$$ If we write $$B_j=\{x:x\in M, x(j)=1\}$$and $A_j=E\cap B_j$ for $j=1,2,\dots$ why is $\tau(A_j)$ a finite union of intervals whose endpoints are of the form $m/2^m$? This is a part of the proof in showing the existence of a Borel isomorphism of the unit interval $I\sim E$ but I don't see how this last claim is easy to see.